\[ \int \frac {\sqrt {b x^2+c x^4}}{x^{15/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}}{11 x^{\frac {13}{2}}}-\frac {4 c \sqrt {c \,x^{4}+b \,x^{2}}}{77 b \,x^{\frac {9}{2}}}+\frac {20 c^{2} \sqrt {c \,x^{4}+b \,x^{2}}}{231 b^{2} x^{\frac {5}{2}}}+\frac {10 c^{\frac {11}{4}} x \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {b}+x \sqrt {c}\right ) \sqrt {\frac {c \,x^{2}+b}{\left (\sqrt {b}+x \sqrt {c}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{b^{\frac {1}{4}}}\right )\right ) b^{\frac {9}{4}} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((c*x^4+b*x^2)^(1/2)/x^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (10 \, c^{\frac {5}{2}} x^{7} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (10 \, c^{2} x^{4} - 6 \, b c x^{2} - 21 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{231 \, b^{2} x^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}}}{x^{\frac {15}{2}}}, x\right ) \]